---

In this paper stability analysis of a nonlinear micro rotating shaft near the primary resonances by considering the modified couple stress theory and micro inertia effect is investigated. The geometric nonlinearities due to classical and non-classical theory (the modified couple stress theory) are considered. Using Hamilton principle, the nonlinear equations of motion are obtained. In order to solve the equations of motion the multiple scales method are used and an analytical expression is presented for forward and backward frequencies which can be seen the effects of modified couple stress theory and micro inertia effect. The frequency response curves, amplitude versus damping coefficient, amplitude versus total eccentricities, etc. are reported. It is seen that due to the modified couple stress theory and micro inertia effect the amplitude of the system is decreased and the loci of bifurcation points is changed. Symmetrical micro-shaft in the presence of classical theory and without micro inertia effects becomes completely stable in the least damping coefficient and asymmetrical micro-shaft in the presence of classical theory and without micro inertia effects becomes completely stable in the most damping coefficient. Symmetrical micro-shaft in the presence of modified couple stress theory and with micro inertia effects becomes completely stable in the least total eccentricity and asymmetrical micro-shaft in the presence of classical theory and without micro inertia effects becomes completely stable in the most total eccentricity. So, considering the small-scale effects due to strain and velocity gradients for analysis of the system is mandatory.

---

In this paper free and forced vibrations analysis of a viscoelastic nonlinear nano rotating beam by considering surface effects is investigated. Using Hamilton principle and Gurtin Murdoch theory, the equations of motion are obtained and discretized by Galerkin method. Using the multiple time scales method the equations of motion are solved. In free vibrations analysis, the analytical expressions for amplitude and phase are obtained. In forced vibrations analysis the steady state solution are obtained. The effect of surface effect, damping coefficients, dimensions of cross section area, external excitation amplitude etc. on frequency response curves are investigated. It is seen that in free vibrations, by increasing surface stress the amplitude of the system decreased, and by increasing surface density or elasticity it is increased. Also, by increasing internal and external damping coefficients free vibration amplitude is decreased. In forced vibrations, it is seen that considering surface effect the amplitude of the system is decreased and the first bifurcation point is obviously changed. By increasing internal and external damping coefficients the amplitude is decreased and the first bifurcation point occur in frequencies near the natural frequency. It is seen that for two different dimensions of cross section with same area, amplitude and the loci of the bifurcation points are changed. By increasing the amplitude of external excitation the amplitude of response is increased the bifurcation points occur in frequencies far away from natural frequency. So, considering the surface effects for free and forced vibrations analysis of the nano rotating beams is mandatory.